Parallels
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From: Greg Neill on 5 May 2010 22:34 Gerry Myerson wrote: > In article <bXfEn.41958$nI2.13354(a)unlimited.newshosting.com>, > "Greg Neill" <gneillRE(a)MOVEsympatico.ca> wrote: > >> William Elliot wrote: >>> What's the definition of parallel lines in Euclidean three space? >>> Two coplanar lines that don't intersect? >> >> That would include skew lines, no? > > No. Skew lines are (by definition) not coplanar. D'oh! Of course.
From: William Elliot on 6 May 2010 06:36 On Wed, 5 May 2010, Hero wrote: > On 5 Mai, 12:10, William Elliot <ma...(a)rdrop.remove.com> wrote: >> What's the definition of parallel lines in Euclidean three space? >> Two coplanar lines that don't intersect? >> >> In Euclidean three space, let the line j be parallel to the line k >> and k parallel to the line l. �Prove j is parallel to l. > > Euclid, Book XI, Proposition 9 > Proposition 9 > Straight lines which are parallel to the same straight line but do not > lie in the same plane with it are also parallel to each other. > > That will cover half of the answer. > If j,k,l lie in the same plane then j and l are parallel by using corresponding angles of a traversal of parallel lines are equal, the sum of angles of a triangle is 180 degrees and lines intersecting at 0 degrees concoincide. What lemmas or theorems are used for proposition 9 above? BTW, why do lines that intersect at 0 degrees conincide?
From: Hero on 6 May 2010 12:25
William wrote: > Hero wrote: > > William wrote: > >> What's the definition of parallel lines in Euclidean three space? > >> Two coplanar lines that don't intersect? > > >> In Euclidean three space, let the line j be parallel to the line k > >> and k parallel to the line l. Prove j is parallel to l. > > > Euclid, Book XI, Proposition 9 > > Proposition 9 > > Straight lines which are parallel to the same straight line but do not > > lie in the same plane with it are also parallel to each other. > > > That will cover half of the answer. > > If j,k,l lie in the same plane then j and l are parallel by > using corresponding angles of a traversal of parallel lines are > equal, the sum of angles of a triangle is 180 degrees and lines > intersecting at 0 degrees concoincide. > > What lemmas or theorems are used for proposition 9 above? > Does this question imply, that You are not familiar with Euclid's Elements ? - I can't believe this, especially because You ask about "Euclidean three space". Anyhow, at this link You find the complete text (as far as we know) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html Now click in Book XI on "Proposition 9" and get Euclids proof of it. And for the propsitions used here one can go backward to read, how they are proofed and so forth - until right to the beginning in Book I Definitions. > BTW, why do lines that intersect at 0 degrees conincide? Not always. Euclid has something nice in Book III, Prop.16. But when You think these lines are straight lines and secondly with Your definition of 0 degrees the answer should not be difficult. With friendly greetings Hero |